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Question 1
State algebraic properties of real numbers.
Question 2
Theorem-: (a) If z and a are elements in with z+a =a, then z=0.
(b) If u and b 0 are elements in with u.b=b,then u=1
(c) If a ,then a.0=0
Question 3
Theorem-: (a) If a 0 and b in such that a.b=1 then b=1/a.
(b) If a.b=0,then either a=o or b=0
Question 4
Theorem-: There does not exist a rational number r such that =2
Definition-: The order properties of
There is a non empty subset P of , called the set of positive real numbers
That satisfies the following properties:
(1) If a, b belong to P,then a+b belongs to P.(2) If a , b belong to P,then ab belong to P.(3) If a belong to ,then exactly one of the following is true:
a P, a=0, -a P
Definition-: let a , b be elements of .
(a) If a-b P ,then we write a>b or b<a
(b) If a-b P ,then we write a b or b a.
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Question 5
Theorem-: let a , b , c be any elements of .
(a) If a>b and b>c , then a>c.(b) If a>b,then a+c >b+c. (c) If a >b and c>0, then ca>cb.
If a> b and c<0, then ca<cb.
Question 6
Theorem-: (a) If a and a 0, then >0.
(b)If n , then n>0.
Question 7
Theorem-: If a is such that 0 a< for every >0, then a=0.
Question 8
Theorem-: If ab>0, then either
(1) a>0 and b>0 or(2) a<0 and b<0
Question 9
(a) Let a o and b 0.Then
a < b
(b) If a and b are positive real numbers, then
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
(c) ,(BERNOULLI’S INEQUALITY).
Question 10
If a , b are real numbers then prove the following.
(a) If a+b=o, then a+b=0 ,then b=-a(b) –(-a)=a(c) (-1)a=-a,(d) (-1)(-1)=1(e) –(a+b)=(-a)+(-b)(f) (–a).(-b)=a.b
(g)
Question 11
If a satisfies a.a=a, prove that either a=o or a=1.
Question 12
,show that.
Question 13
Show that there does not exist a rational number s such that =6
Question 14
Show that there is no rational whose square is 3.
Question 15
Show that sum and product of two rational numbers, is a rational number.
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Question 16
(a) Prove that if x is rational and y is irrational then x+y is irrational.(b) If x is non zero rational and y is irrational then x.y is irrational.(c) Show that there is no natural number between 0 and 1(d) Prove that no natural number can be both even and odd
(e) If c>1 and m , n are natural numbers ,show that .
(f) If 0<c<1 and m , n are natural numbers,show that
Question 17
(a) If a<b and c d, prove that a+c<b+d.(b) If 0<a<b and 0 c d,prove that 0 a.c b.d.
(c) Show that if a>0 , then >0 and .
(d) Show that if a<b , then .
Question 18
(a) If 0<c<1,show that 0<
(b) If 1<c, show that .
(c) If
Absolute value and real line
Definition-:The absolute value of a real number a, denoted by ,is defined by
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Question 19
Theorem- : (a)
Question 20
Theorem-:(Triangle inequality)
If .
Question 21
Corollary-: If
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Question 22
Definition-:
Question 23
Theorem-:
.
Question 24
Question 25
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Question 26
Qestion 27
.
Question 28
.
Question 29
Question 30
Question 31
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Question 32
Question 33
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385
Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385